I follow your argument as to how the central hole is the only reasonable end point for the final ball.
However, your article promises to show how group theory can devise a strategy for solving the puzzle.
It does not do this.
I am sure that group theory really does have applications in chemistry but I have yet to see concrete examples.
Unfortunately, all the articles I read on this topic, including yours, merely give examples of symmetry in compounds and how this can be represented in terms of symmetry groups.
They go on to refer to how it is useful in interpreting IR and other spectroscopy but do not show how.
I was using spectra to deduce structures 60 years ago. The effect of chirality on polarized light has been known for a century.
And I hadn't heard of GT until last week.
I would be grateful if you could point me in the direction of articles showing how group theory adds to this.